proving a matrix is symmetric

Question

Let $A$ be an $nxn $ symmetric matrix.

a) Show that $A^2 $ is symmetric.

b) Show that $2A^2 -3A + I$ is symmetric.

for part a), i have:

$A=A^T$

$A^2 = A\times A$

$A^2 = (A^T)\times(A^T)$

$A^2 = (A\times A)^T$

$A^2 = (A^2)^T$

since $A^2 $is transposable, it is symmetric.

part b) i can prove similar as part a).

Any correction?

Answer 1

In part b) you will need the fact $(A+B)^T=A^T+B^T$ and $(3A)^T=3A^T$and the same idea solves the problem.